Transcript Chapter 13
Chapter 13
Section 13.1
Rectangular Space Coordinates
Coordinates in 3 Dimensions (Space) In 3 dimensions points have 3 coordinates 𝑥, 𝑦, 𝑧 located along 3 different axis:
x
-axis goes back to front
y
-axis goes left to right
z
-axis goes bottom to top
When we draw this we show a projected view (off center), but it is really the case the
x
,
y
and
z
axis meet at right angles 𝑃: 2, −1,3
x z M
𝑄: 4,6,5
y
Midpoint of the points 𝑃: 𝑥 1 , 𝑦 1 , 𝑧 1 and 𝑄: 𝑥 2 In this example the midpoint
M
of the points
P
, 𝑦 2 , 𝑧 2 and
Q
is given by: 𝑥 1 +𝑥 2 2 , 𝑦 1 +𝑦 2 2 , 𝑧 1 +𝑧 2 2 is: 2+4 2 , −1+6 2 , 3+5 2 = 3, 5 2 , 4
z
Distance formula The distance between the points 𝑃: 𝑥 1 , 𝑦 1 , 𝑧 1 and 𝑄: 𝑥 2 , 𝑦 2 , 𝑧 2 is given by: 𝑥 2 − 𝑥 1 2 + 𝑦 2 − 𝑦 1 2 + 𝑧 2 − 𝑧 1 2 𝑥 2 − 𝑥 1 2 + 𝑦 2 − 𝑦 1 2 + 𝑧 2 − 𝑧 1 2 𝑥 1 , 𝑦 1 , 𝑧 1 𝑥 2 , 𝑦 2 , 𝑧 2 𝑧 2 − 𝑧 1
y
In this example: 4 − 2 2 + 6 − −1 2 + 5 − 3 2 = 4 + 49 + 4 = 57
x
𝑥 2 − 𝑥 1 2 + 𝑦 2 − 𝑦 1 2
Equation of a Sphere A sphere with center point 𝐶 𝑥 0 , 𝑦 0 , 𝑧 0
r
has equation: and radius of 𝑥 − 𝑥 0 2 + 𝑦 − 𝑦 0 2 + 𝑧 − 𝑧 0 2 = 𝑟 2
C r
𝑥, 𝑦, 𝑧 Example Find the equation of the sphere whose diameter has endpoints 5, −4,9 and 3,2, −9 .
The center of the sphere is the midpoint of the diameter. 5+3 2 , −4+2 2 , 9+ −9 2 = 4, −1,0 The radius of the sphere is the distance from the center that we just found to one point.
𝑟 = 5 − 4 2 + −4 − −1 2 + 9 − 0 2 = 1 + 9 + 81 = 91 The equation is: 𝑥 − 4 2 + 𝑦 + 1 2 + 𝑧 2 = 91 What is the equation of: 𝑦 = −1 − 91 − 𝑥 − 4 2 − 𝑧 2 Since we solved the equation of the sphere for
y
and took the negative root this is the left half of a sphere. (Sometimes called a hemisphere)
Coordinate planes The graph formed by setting each variable to zero is the coordinate plane of the other two variables.
xy
plane 𝑧 = 0
xz
plane 𝑦 = 0
yz
plane 𝑥 = 0
x z
𝑧 = 0
y x z
𝑦 = 0
y x z
𝑥 = 0
y
𝑥 = −5 Changing what values that
x
,
y
and
z
are being set equal gives a plane that is parallel to one of the coordinate planes
x
Lines in 3 Dimensions A line in 3 dimensions is described using 3 equations each giving the
x
,
y
and
z
coordinates on the line with an independent variable called a parameter.
z
𝑧 = 1
y
𝑧 = 0 𝑦 = 0
x
Parametric equations for a line passing through the points
z
𝑎, 𝑏, 𝑐 𝑦 = 3 and
y x
𝑑, 𝑒, 𝑓 .
𝑥 = 𝑑 − 𝑎 𝑡 + 𝑎 𝑦 = 𝑒 − 𝑏 𝑡 + 𝑏 𝑧 = 𝑓 − 𝑐 𝑡 + 𝑐
z
𝑥 = 0
y